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<title>Atlas software user guide -- Inner classes</title>
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<h2>Inner classes</h2>
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<i>Last updated: October 8, 2005</i>
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Let G be a connected reductive complex algebraic group. The <i>inner class</i> 
of a <a href="realforms.html">real form</a> of G is its image in the outer 
automorphism group of G. We look at inner classes up to conjugacy in Out(G); 
it may be shown that there are always finitely many possiblities.
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Let Rad(G) be the identity component of the center of G. Then an inner class
is entirely determined by the involutions it induces on the Dynkin diagram of
G, and on Rad(G). If G is the direct product of Rad(G) and the derived group
Der(G), and if Der(G) is simply connected, or adjoint, all pairs of involutions
are allowed; this is no longer true in general (the simplest case is when
G is PSL(2).SL(2); then the complex inner form (see below) is <i>not</i> 
allowed.)
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The involution induced on the Dynkin diagram of G either fixes a component, or
interchanges it with an isomorphic one. We require that the pairs of 
interchanged components correspond to consecutive entries in the Lie type
(for instance, it is allowed to interchange the two A1's in B3.A1.A1, but
not in A1.B3.A1.) Of course, it is always possible to lay out the group G in
such a way that this condition is fulfilled. We now have a number of 
possibilities:
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A pair of interchanged components gives rise to a <i>complex</i> factor (i.e.,
a complex reductive algebraic group viewed as real);
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A fixed component on which the involution induces the identity will be labelled
&#8220;c&#8221; (for compact), or also &#8220;e&#8221; (for equal rank). This 
corresponds to the fact that the inner class induced on that factor will be the
one containing the compact real form, also called the <i>equal rank</i> inner 
class;
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For a fixed component of one of the types A_n, n >= 2, D_n, n odd, and E6, the 
non-identity involution of the Dynkin diagram is denoted &#8220;s&#8221;, 
because the inner class induced on that factor will be the one containing the 
split real form; this may also be denoted &#8220;u&#8221; (for unequal rank.) 
For the types A1, B_n, C_n, E7, E8, F4, G2, there is a unique 
inner class, which may be denoted either &#8220;c&#8221; or &#8220;s&#8221;;

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For a fixed component of <a href="Deven.html">type D_2m</a>, the non-identity
involution of the Dynkin diagram is denoted &#8220;u&#8221;;
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As recalled <a href="tori.html">here</a>, any torus with involution can be 
split up into a direct product of factors on which the involution is either 
the identity (labelled &#8220;c&#8221;as above), inversion (labelled 
&#8220;s&#8221;), or the exchange of two isomorphic torus factors (labelled 
&#8220;C&#8221; for complex). By choosing the way we write Rad(G) 
appropriately, we may therefore assume that the involution is either the 
identity, the inversion, or the exchange of two consecutive isomorphic torus 
factors on the various factors of Rad(G).
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The upshot is that without loss of generality we may represent our inner class
by a sequence of symbols from the set {c,s,C,u}, one for each factor in the
Lie type, except that a &#8220;C&#8221; symbol uses up two consecutive 
isomorphic factors, and with the restriction that "u" is allowed only when 
there is more than one inner class. If it turns out that a given sequence is 
not allowed for the specific covering group G that we are considering, the 
program will notice, and complain accordingly.
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<p>
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